However, although the diffeomorphism argument does not succeed, there may be other reasons to conclude that we should think of the two branches as containing `the same events.' To get clearer on this question, it is helpful to consider in more detail what is actually achieved by the quantum SWITCH. On the one hand, the results of refs [1]}, [2]} underline that the quantum SWITCH does not exhibit the strongest and most interesting kind of `indefinite causal order' that is possible within the process matrix frameworkRef [1]} demonstrates that the quantum SWITCH is not equivalent to the gravitational SWITCH, which implements a similar circuit but uses superpositions of different spacetime geometries to determine the two different orders; this indicates that the ordinary quantum SWITCH does not exhibit indefinite causal order in the same unambiguous way as the gravitational SWITCH. Meanwhile, the results of [2]} show that at the purely operational level it would be possible to come up with a `fine-grained' version of the quantum SWITCH process which can be assigned a well-defined causal order, so operationally speaking the SWITCH is not achieving anything that could not be achieved with a fixed causal order.. On the other hand, if Maclean et al were right that in fact completely different events occur in each branch, then one would naturally expect the quantum SWITCH to be equivalent to a superposition in which the qubit goes down two different paths, one containing the gate \(A\)  and the gate \(B\) , and another containing a completely distinct copy of the gate \(B\)  and then a completely distinct copy of the gate \(A\) . However, as shown in ref [5]}, the quantum SWITCH is not equivalent to this superposition - rather it is equivalent to a superposition in which the two paths are correlated. That is to say, the quantum SWITCH produces the same results as a scenario where we have one spacetime region containing a system \(S_A\)  which implements gate \(A\)  and another spacetime region containing a system \(S_B\)  which implements gate \(B\) , and in one branch the qubit first visits the \(A\) -region and then the \(B\) -region, while in the other branch the qubit first visits the \(B\) -region and then the \(A\) -region. In this case it is easy to see that the reason the paths become correlated is precisely because the copy of gate A that the qubit visits is really the same gate in each branch: in the branch where the qubit visits the \(A\) -region first, it interacts with \(S_A\)  and undergoes gate \(A\) , and meanwhile the state of \(S_A\)  is presumably altered, but in the other branch nothing happens during that period of time and instead the qubit visits the \(A\)  region later, finding \(S_A\)  still in its original quantum state so that gate \(A\)  is implemented. So this equivalence seems to uphold the conclusion of Guérin and Brukner that the quantum SWITCH is in some sense a superposition of different causal orders.
